For any \(v\) -stack \(X\) , let \(D_{\mathop {\mathrm {\acute{e}t}}}(X,\mathbf {Z}/\ell ^n) \subset D(X_v, \mathbf {Z}/\ell ^n)\)  be the full subcategory spanned by objects whose pullback along any map \(Y \rightarrow X\)  with \(Y\)  a strictly totally disconnected perfectoid space lies in the subcategory \(D(Y_{\mathop {\mathrm {\acute{e}t}}}, \mathbf {Z}/\ell ^n) \subset D(Y_v,\mathbf {Z}/\ell ^n)\) ; this condition can be checked after pullback along a \(v\) -cover ([1]}. Imposing this condition for a complex \(K\)  is equivalent to imposing it for each \(v\) -cohomology sheaf \(\mathcal {H}^i(K)\)  (regarded as a complex) ([1]}). Moreover, if \(X\)  is a locally spatial diamond (e.g., one attached to a rigid space), then \(D_{\mathop {\mathrm {\acute{e}t}}}(X,\mathbf {Z}/\ell ^n)\)  admits a classical description: it agrees with the left-completion \(\widehat{D}(X_{et},\mathbf {Z}/\ell ^n)\)  of the usual derived category \(D(X_{\mathop {\mathrm {\acute{e}t}}},\mathbf {Z}/\ell ^n)\)  ([1]}).
